A \emph{signature}$\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow\nat$. Abusing notations, we will often write $R\in\ssign$ instead of $R\in S$.
\paragraph{Structures} A \emph{structure}$A$ over a signature $\ssign$ is given as a \emph{domain}$\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)-\subseteq\dom(A)^r$, with $r=\ar(R)$.
\paragraph{Structures} A \emph{structure}(sometimes \emph{model}) $A$ over a signature $\ssign$ is given as a \emph{domain}$\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)\subseteq\dom(A)^r$, with $r=\ar(R)$.
\paragraph{Operations}
Any structure over a signature can be seen as a structure over a larger signature where additional symbols are interpreted as empty sets.
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@@ -107,9 +113,9 @@ The \emph{product} of two structures $A,B$ over a signature $\ssign$ is a struct
The \emph{ordered coproduct} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigoplus_{i\leq n} A_i$ over the signature $\ssign+\set{\sqsubseteq}$, where $\sqsubseteq$ has arity $2$. The domain of $\bigoplus_{i\leq n} A_i$ is $\bigoplus_{i\leq n}\dom_{A_i}=\bigcup_{i\leq n}\dom_{A_i}{\times}\set{i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n}\inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^1+\ssign^{>1}\times\set{\sqsubseteq}$. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n}\dom_{A_i}$.
The interpretation of a symbol $R$ is $\bigoplus_{i\leq n}\inter_{A_i}(R)$. The interpretation of $\sumorder$ is$\bigcup_{i\leqj\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^{1}+\ssign^{>1}\times\set{\sqsubseteq}$, where $\ssign^{1}$, $\ssign^{>1}$ denote the symbols of arity one and at least two, respectively. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n}\dom_{A_i}$.
The interpretation of a symbol $R$of arity $1$is $\bigotimes_{i\leq n}\inter_{A_i}(R)$. The interpretation of $(R,\sumorder)$ is: $$\bigcup_{j\leq n}\quad\bigotimes_{i<j}\Delta_{i,k}\times\inter_{A_j}\times\bigotimes_{j<i\leq n}\dom_{A_i}$$
where $\Delta_{i,k}=\set{(a,\ldots,a)\in\dom_{A_i}^k\mid\ a\in\dom_{A_i}}$ is the $k$-fold diagonal of $\dom_{A_i}$.
The \emph{powerset structure} of a structure $A$ over signature $\ssign$ is a structure $\pow A$ over signature $\ssign\uplus\set{\subseteq}$, where $\subseteq$ is a binary symbol. The domain of $\pow A$ is $\pow{\dom_A}$. The interpretation of a symbol $R$ is the set $\set{(\set{x_1},\dots,\set{x_k})\mid\ (x_1,\ldots,x_n)\in\inter_A(R)}$.
The interpretation of $\subseteq$ is the actual set inclusion $\set{(x,y)\mid\ x\subseteq y\subseteq\dom_A}$.
\paragraph{Word models}
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@@ -234,26 +240,6 @@ An \msomi of dimension $d$ has growth $O((2^n)^d)=O(2^{dn})=2^{O(n)}$.
\section{The case of the successor}
\begin{theorem}
\msomi with successor is equivalent to \dlin reductions.
\end{theorem}
\begin{proof}
Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton.
From a linear bounded automaton, the next configuration relation can be defined in \mso.
\end{proof}
\begin{question}
Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ\cdots\circ f_{a_n}(\epsilon)$ compute \dlin ?
\end{question}
\section{Nested marble transducer}
We define a model of transducers based on marble transducers.
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@@ -299,7 +285,7 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give
\end{remark}
\begin{example}
Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup\set{\sharp}\right)^*\rightarrow(A\times\cup{0,1}\cup\set{\natural})^*$.
Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup\set{\sharp}\right)^*\rightarrow(A\times\cup\set{0,1,\natural})^*$.
What the function $f$ does is list all subsets of positions (without $\sharp$) and output the word labelled with the corresponding subset. The labelled words are output in a specific order, and separated by $\natural$. Let us decribe the order in which these subsets of positions are listed.
Over a word $u$ without $\sharp$, $f$ outputs the list of all annotations, in the lexicographic order over the subset of positions of $u$, \eg
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@@ -352,8 +338,35 @@ Does $\ipt\subseteq \mt\circ\msot$ hold?
\end{question}
\section{Rational Turing Machine}
A \emph{Rational Turing Machine} is a Turing Machine with a reachability relation over configurations that is rational.
\begin{theorem}
A function is \expreg if and only if it is definable by a \drtm.
\end{theorem}
\begin{theorem}
For rational turing machines, the following hold:
\begin{enumerate}
\item\drtm = \dlinspace
\item\dlogspace= \dptime
\end{enumerate}
\end{theorem}
\section{The case of the successor}
\begin{theorem}
\msomi with successor is equivalent to \dlinspace reductions.
\end{theorem}
\begin{proof}
Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton.
From a linear bounded automaton, the next configuration relation can be defined in \mso.
\end{proof}
\section{Some remaining questions}
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@@ -386,6 +399,8 @@ Given a (fixed) \msomi realizing $f$, can one compute the image of a word $u$ in
\item recursive programming language corresponding to or being
captured by \msomi ?
\item Krohn-Rhodes like decomposition \eg$\msoi\circ\mathsf{exp}\circ\msot$, with $\mathsf{exp}$ being some simple class of exponential growth function.
\item Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ\cdots\circ f_{a_n}(\epsilon)$, with sequential functions compute \dlinspace ?
